Sign of effective action of bosonic string?

Question

I've been looking at David Tong's Lectures on String Theory.

He states that the low-energy effective action of the bosonic string is given by $$S=\frac{1}{2k_0^2}\int d^{26}X\sqrt{-G}e^{-2\Phi}\Big(\mathcal{R}-\frac{1}{12}H_{\mu\nu\lambda}H^{\mu\nu\lambda}+4\partial_\mu\Phi\partial^\mu\Phi\Big)\tag{7.16}$$

If this expression is related to the Polyakov action $$S=-\frac{1}{4\pi\alpha'}\int d^2\sigma\sqrt{-g}g^{\alpha\beta}\partial_\alpha X^\mu\partial_\beta X^\nu \eta_{\mu\nu}\tag{1.22}$$

then should it also have a minus sign at the front?

I ask this because if there is a minus sign in front of (7.16) then the kinetic and potential terms of the dilaton field $\Phi$, including the interaction term with the Ricci scalar, have the correct sign for a scalar lagrangian given David Tong's choice of signature $(-1,+1,+1,...,+1)$.

One can argue that the overall sign of the action doesn't matter but in the second paragraph of section 7.3.1 Tong himself worries about the sign of the kinetic term for $\Phi$. Is he wrong to be worried?

Answer 1

The two signs are not correlated.

Also: your (Tong's) (7.16) is not in a canonical form where you can immediately determine the positivity of the energy by reading off the sign in the kinetic term. (This is why we make people have positive $\dot\phi^2$ terms in their lagrangians, although as @ACuriousMind mentions the actual sign is arbitrary; it is only important for the energy of a system to have either an upper bound or a lower bound.) And indeed Tong claims (but I didn't check myself) that when you do bring this lagrangian to canonical form by going to Einstein frame, all the kinetic terms are positive.